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If the tens digit of the positive integer x and the tens digit of the

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Bunuel
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If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   03 Jan 2024, 06:46
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If the tens digit of the positive integer x and the tens digit of the positive integer y are both 6, what is the number of different possible values for the tens digit of 2x + 2y?

A. Three
B. Four
C. Five
D. Six
E. Seven

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B
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chetan2u
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   03 Jan 2024, 08:00
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Bunuel
If the tens digit of the positive integer x and the tens digit of the positive integer y are both 6, what is the number of different possible values for the tens digit of 2x + 2y?

A. Three
B. Four
C. Five
D. Six
E. Seven

Show Spoiler
Attachment:
Screenshot 2024-01-02 202125.png

The tens digit of 2x+2y will depend only on the units digit, so it doesn’t matter what other digits are.
Let the two numbers be 6a and 6b.
Least value of 2(x+y) = 2(60+60) = 240
Maximum value of 2(x+y) = 2(69+69) = 276

So 2(x+y) can take all even values from 240 to 276.
Thus, tens digit can be 4, 5, 6 and 7, a total of four values.

B
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gmatophobia
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   03 Jan 2024, 22:50
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Bunuel
If the tens digit of the positive integer x and the tens digit of the positive integer y are both 6, what is the number of different possible values for the tens digit of 2x + 2y?

A. Three
B. Four
C. Five
D. Six
E. Seven

Show Spoiler
Attachment:
Screenshot 2024-01-02 202125.png

\(x = 6p\) ⇒ \(60 + p\)

\(y = 6q\) ⇒ \(60 + q\)

\(2x = 120 + 2p\)

\(2y = 120 + 2q\)

\(2x + 2y = 240 + 2(p+q)\)

The minimum value of \(p + q = 0\), and the maximum value of \(p + q = 18\)

Hence, the minimum value of \(2(p+q) = 0\), the maximum value of \(2(p+q) = 36\). From this information, we can infer that while performing the addition operation we can have a carryover of 0, 1, 2, or 3 from the unit place to the tens place.

Possible values of the tens place = (4+0), (4+1), (4+2), (4+3) = 4, 5, 6, or 7.

Option B
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   05 Jan 2024, 04:57
gmatophobia
Bunuel
If the tens digit of the positive integer x and the tens digit of the positive integer y are both 6, what is the number of different possible values for the tens digit of 2x + 2y?

A. Three
B. Four
C. Five
D. Six
E. Seven

Show Spoiler
Attachment:
Screenshot 2024-01-02 202125.png

\(x = 6p\) ⇒ \(60 + p\)

\(y = 6q\) ⇒ \(60 + q\)

\(2x = 120 + 2p\)

\(2y = 120 + 2q\)

\(2x + 2y = 240 + 2(p+q)\)

The minimum value of \(p + q = 0\), and the maximum value of \(p + q = 18\)

Hence, the minimum value of \(2(p+q) = 0\), the maximum value of \(2(p+q) = 36\). From this information, we can infer that while performing the addition operation we can have a carryover of 0, 1, 2, or 3 from the unit place to the tens place.

Possible values of the tens place = (4+0), (4+1), (4+2), (4+3) = 4, 5, 6, or 7.

Option B

and the maximum value of p+q=18
why's this the case please?
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gmatophobia
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   05 Jan 2024, 05:45
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tickledpink001
and the maximum value of p+q=18
why's this the case please?

tickledpink001 - \(p\) and \(q\) are single digits non-negative integers (i.e. both \(p\) and \(q\) can take values from \(0\) to \(9\), inclusive). Hence, the maximum value that \(p\) and \(q\) can have is \(9\), resulting in the maximum value of \(p + q = 18\).

Hope this helps.
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   15 Mar 2024, 20:07
We have to find the unique tens digit number from 2x + 2y

Min number could be 61 and max number could be 69.

So the range of numbers from 240 to 276 which we have to find the unique number of tens digits.

Only unique digits we will get from range 240 to 276 is 4 , 5, 6, 7 as unique tens digits

So total 4 unique digits.

Option - B will be the answer
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   27 Apr 2024, 13:55
chetan2u
Bunuel
If the tens digit of the positive integer x and the tens digit of the positive integer y are both 6, what is the number of different possible values for the tens digit of 2x + 2y?

A. Three
B. Four
C. Five
D. Six
E. Seven

Show Spoiler
Attachment:
Screenshot 2024-01-02 202125.png
The tens digit of 2x+2y will depend only on the units digit, so it doesn’t matter what other digits are.
Let the two numbers be 6a and 6b.
Least value of 2(x+y) = 2(60+60) = 240
Maximum value of 2(x+y) = 2(69+69) = 276

So 2(x+y) can take all even values from 240 to 276.
Thus, tens digit can be 4, 5, 6 and 7, a total of four values.

B
­Why did you assume the integer is a 2 digit integer? Thank you.
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   27 Apr 2024, 19:37
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Engineer1
chetan2u
Bunuel
If the tens digit of the positive integer x and the tens digit of the positive integer y are both 6, what is the number of different possible values for the tens digit of 2x + 2y?

A. Three
B. Four
C. Five
D. Six
E. Seven

Show Spoiler
Attachment:
Screenshot 2024-01-02 202125.png
The tens digit of 2x+2y will depend only on the units digit, so it doesn’t matter what other digits are.
Let the two numbers be 6a and 6b.
Least value of 2(x+y) = 2(60+60) = 240
Maximum value of 2(x+y) = 2(69+69) = 276

So 2(x+y) can take all even values from 240 to 276.
Thus, tens digit can be 4, 5, 6 and 7, a total of four values.

B
­Why did you assume the integer is a 2 digit integer? Thank you.
­We don't know the number of digits, but it doesn't matter. When we add, no digit affects the value of those to its right, only those to its left. 
For instance, if we add 2169 + 53,869, the answer still ends in 38, just as when we add 69 + 69. Similarly, when we double the value, those additional places to the left don't change the fact that 38 doubles to 76. 

So in short, we can focus only on the relevant digits and not worry about any other potential digits. This is common on place value/digit problems.
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   28 Apr 2024, 07:36
 
DmitryFarber
­We don't know the number of digits, but it doesn't matter. When we add, no digit affects the value of those to its right, only those to its left. 
For instance, if we add 2169 + 53,869, the answer still ends in 38, just as when we add 69 + 69. Similarly, when we double the value, those additional places to the left don't change the fact that 38 doubles to 76. 

So in short, we can focus only on the relevant digits and not worry about any other potential digits. This is common on place value/digit problems.
­Understood, thank you. I think you meant this, correct?  "When we add, no digit affects the value of those to its left, only those to its right."­
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   28 Apr 2024, 18:54
chetan2u
Bunuel
If the tens digit of the positive integer x and the tens digit of the positive integer y are both 6, what is the number of different possible values for the tens digit of 2x + 2y?

A. Three
B. Four
C. Five
D. Six
E. Seven

Show Spoiler
Attachment:
Screenshot 2024-01-02 202125.png
The tens digit of 2x+2y will depend only on the units digit, so it doesn’t matter what other digits are.
Let the two numbers be 6a and 6b.
Least value of 2(x+y) = 2(60+60) = 240
Maximum value of 2(x+y) = 2(69+69) = 276

So 2(x+y) can take all even values from 240 to 276.
Thus, tens digit can be 4, 5, 6 and 7, a total of four values.

B
­What do you mean by even values? Also  why is the assumption that the "positive" integer for x and y is only 2 digits (and not bigger)? Help is appreciated :)
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chetan2u
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   28 Apr 2024, 19:07
Expert Reply
Danou
chetan2u
Bunuel
If the tens digit of the positive integer x and the tens digit of the positive integer y are both 6, what is the number of different possible values for the tens digit of 2x + 2y?

A. Three
B. Four
C. Five
D. Six
E. Seven

Show Spoiler
Attachment:
Screenshot 2024-01-02 202125.png
The tens digit of 2x+2y will depend only on the units digit, so it doesn’t matter what other digits are.
Let the two numbers be 6a and 6b.
Least value of 2(x+y) = 2(60+60) = 240
Maximum value of 2(x+y) = 2(69+69) = 276

So 2(x+y) can take all even values from 240 to 276.
Thus, tens digit can be 4, 5, 6 and 7, a total of four values.

B
­What do you mean by even values? Also  why is the assumption that the "positive" integer for x and y is only 2 digits (and not bigger)? Help is appreciated :)

Even values meant even integers between 240 and 276.

Next if the numbers are 768964 and 876543865, we are looking for TENS digit of 2(768964 + 876543865). TENS digit will be given only by ones digit and tens digit, so why waste time on thinking what other digits are => 2(64+65)

Posted from my mobile device
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink] If the tens digit of the positive integer x and the tens digit of the   23 Jun 2024, 03:53
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Bunuel
If the tens digit of the positive integer x and the tens digit of the positive integer y are both 6, what is the number of different possible values for the tens digit of 2x + 2y?

A. Three
B. Four
C. Five
D. Six
E. Seven

Show Spoiler
Attachment:
Screenshot 2024-01-02 202125.png
­x and y both are of the form ...60 or ...61 or ... or ...69 

When we add them (x + y), the sum ranges from
...60 + ...60 = ...20
...60 + ...61 = ...21
to
...68 + ...69 = ...37
...69 + ...69 = ...38

Last two digits will be anything from 20 to 38. 
When we multiply this sum by 2 to get 2(x+y), the value ranges from 

2 * ...20 = ...40
2* ...21 = ...42
to
2*...37 = ...74
2*...38 = ...76

Hence the tens digit of 2(x+y) can be anything from 4 to 7 i.e. it can take 4 distinct values. 

Answer (B)
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Re: If the tens digit of the positive integer x and the tens digit of the [#permalink]
23 Jun 2024, 03:53
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